The above massive gauge-invariant actions are manifestly supersymmetric,
that is the \({\cal N}=1\)  AdS supersymmetry is realised off-shell.
It is worth pointing out that there also exists an on-shell construction of gauge-invariant Lagrangian formulations for massive higher-spin \({{\cal N}}=1\)  supermultiplets in \(\mathbb {R}^{2,1}\)  and AdS\(_{3}\)  [1]}, [2]}, [3]}, extending previous works on the non-supersymmetric cases [4]}, [5]}. These frame-like formulations are based on the gauge-invariant approach to the dynamics of massive higher-spin fields proposed by Zinoviev [6]}, [7]} and Metsaev [8]}.
